{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Doppler Broadening"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<i>© Von P. Walden, Washington State University</i>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Calculation of the Doppler line shape, caused by relative velocities within a gas."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Populating the interactive namespace from numpy and matplotlib\n"
     ]
    }
   ],
   "source": [
    "%pylab inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "v0 = 31.26e12            # 1042 cm-1 (for Ozone).\n",
    "kB = 1.3806488e-23       # Boltzmann constant\n",
    "m  = (3.*16) / 6.02e23   # Molar mass divided by Avogadro's number\n",
    "c2 = (3e8)**2            # Speed of light squared.\n",
    "T  = 300.               # Kelvin\n",
    "aD = v0 * (2.*kB*T/(m*c2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.03608578757746667"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "aD"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "vij = 1042.            # cm-1\n",
    "c   = 2.99792458e10    # cm s-1\n",
    "Na  = 6.02214129e23    # mo1-1\n",
    "k   = 1.3806488e-16    # erg K-1\n",
    "M   = 48.00            # Molar mass of ozone in grams mo1-1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.0009328929976518732\n"
     ]
    }
   ],
   "source": [
    "aD = (vij / c) * (2.*Na*k*T*log(2)/M)**0.5\n",
    "print(aD)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0.5, 1.0, 'Doppler Line Shape (Doppler broadening)')"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "varr = arange(-0.1,0.1,0.001)\n",
    "fD = 1./(aD*pi**0.5)*exp(-varr**2 / aD**2)\n",
    "\n",
    "plot(varr/aD,fD)\n",
    "xlabel('(v-v0)')\n",
    "ylabel('Normalized f(v-v0)')\n",
    "title('Doppler Line Shape (Doppler broadening)')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x10e304450>"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#legend(('288 K','217 K','271 K','236 K'),loc='best')"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.3"
  },
  "widgets": {
   "state": {},
   "version": "1.1.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}
